Minimum energy control for in vitro neurons

Developing control-theoretic objectives for large-scale brain dynamics and cognitive enhancement

The development of technologies for brain stimulation provides a means for scientists and clinicians to directly actuate the brain and nervous system. Brain stimulation has shown intriguing potential in terms of modifying particular symptom clusters in patients and behavioral characteristics of subjects. The stage is thus set for optimization of these techniques and the pursuit of more nuanced stimulation objectives, including the modification of complex cognitive functions such as memory and attention. Control theory and engineering will play a key role in the development of these methods, guiding computational and algorithmic strategies for stimulation. In particular, realizing this goal will require new development of frameworks that allow for controlling not only brain activity, but also latent dynamics that underlie neural computation and information processing. In the current opinion, we review recent progress in brain stimulation and outline challenges and potential research pathways associated with exogenous control of cognitive function.

Data-driven phase-isostable reduction for optimal nonfeedback stabilization of cardiac alternans

Phase-isostable reduction is an emerging model reduction strategy that can be used to accurately replicate nonlinear behaviors in systems for which standard phase reduction techniques fail. In this work, we derive relationships between the cycle-to-cycle variance of the reduced isostable coordinates for systems subject to both additive white noise and periodic stimulation. Using this information, we propose a data-driven technique for inferring nonlinear terms of the phase-isostable coordinate reduction framework. We apply the proposed model inference strategy to the biologically motivated problem of eliminating cardiac alternans, an arrhythmia that is widely considered to be a precursor to more deadly cardiac arrhythmias. Using this strategy, by simply measuring a series of action potential durations in response to periodic stimulation, we are able to identify energy-optimal, nonfeedback control inputs to stabilize a period-1, alternans-free solution.

Analysis of input-induced oscillations using the isostable coordinate framework

Many reduced order modeling techniques for oscillatory dynamical systems are only applicable when the underlying system admits a stable periodic orbit in the absence of input. By contrast, very few reduction frameworks can be applied when the oscillations themselves are induced by coupling or other exogenous inputs. In this work, the behavior of such input-induced oscillations is considered. By leveraging the isostable coordinate framework, a high-accuracy reduced set of equations can be identified and used to predict coupling-induced bifurcations that precipitate stable oscillations. Subsequent analysis is performed to predict the steady state phase-locking relationships. Input-induced oscillations are considered for two classes of coupled dynamical systems. For the first, stable fixed points of systems with parameters near Hopf bifurcations are considered so that the salient dynamical features can be captured using an asymptotic expansion of the isostable coordinate dynamics. For the second, an adaptive phase-amplitude reduction framework is used to analyze input-induced oscillations that emerge in excitable systems. Examples with relevance to circadian and neural physiology are provided that highlight the utility of the proposed techniques.

First-order synchronization transition in a large population of strongly coupled relaxation oscillators

Onset and loss of synchronization in coupled oscillators are of fundamental importance in understanding emergent behavior in natural and man-made systems, which range from neural networks to power grids. We report on experiments with hundreds of strongly coupled photochemical relaxation oscillators that exhibit a discontinuous synchronization transition with hysteresis, as opposed to the paradigmatic continuous transition expected from the widely used weak coupling theory. The resulting first-order transition is robust with respect to changes in network connectivity and natural frequency distribution. This allows us to identify the relaxation character of the oscillators as the essential parameter that determines the nature of the synchronization transition. We further support this hypothesis by revealing the mechanism of the transition, which cannot be accounted for by standard phase reduction techniques.

Entrainment of a network of interacting neurons with minimum stimulating charge

Periodic pulse train stimulation is generically used to study the function of the nervous system and to counteract disease-related neuronal activity, e.g., collective periodic neuronal oscillations. The efficient control of neuronal dynamics without compromising brain tissue is key to research and clinical purposes. We here adapt the minimum charge control theory, recently developed for a single neuron, to a network of interacting neurons exhibiting collective periodic oscillations. We present a general expression for the optimal waveform, which provides an entrainment of a neural network to the stimulation frequency with a minimum absolute value of the stimulating current. As in the case of a single neuron, the optimal waveform is of bang-off-bang type, but its parameters are now determined by the parameters of the effective phase response curve of the entire network, rather than of a single neuron. The theoretical results are confirmed by three specific examples: two small-scale networks of FitzHugh-Nagumo neurons with synaptic and electric couplings, as well as a large-scale network of synaptically coupled quadratic integrate-and-fire neurons.

A data-driven phase and isostable reduced modeling framework for oscillatory dynamical systems

Phase-amplitude reduction is of growing interest as a strategy for the reduction and analysis of oscillatory dynamical systems. Augmentation of the widely studied phase reduction with amplitude coordinates can be used to characterize transient behavior in directions transverse to a limit cycle to give a richer description of the dynamical behavior. Various definitions for amplitude coordinates have been suggested, but none are particularly well suited for implementation in experimental systems where output recordings are readily available but the underlying equations are typically unknown. In this work, a reduction framework is developed for inferring a phase-amplitude reduced model using only the observed model output from an arbitrarily high-dimensional system. This framework employs a proper orthogonal reduction strategy to identify important features of the transient decay of solutions to the limit cycle. These features are explicitly related to previously developed phase and isostable coordinates and used to define so-called data-driven phase and isostable coordinates that are valid in the entire basin of attraction of a limit cycle. The utility of this reduction strategy is illustrated in examples related to neural physiology and is used to implement an optimal control strategy that would otherwise be computationally intractable. The proposed data-driven phase and isostable coordinate system and associated reduced modeling framework represent a useful tool for the study of nonlinear dynamical systems in situations where the underlying dynamical equations are unknown and in particularly high-dimensional or complicated numerical systems for which standard phase-amplitude reduction techniques are not computationally feasible.

Controlling Synchronization of Spiking Neuronal Networks by Harnessing Synaptic Plasticity

Disrupting the pathological synchronous firing patterns of neurons with high frequency stimulation is a common treatment for Parkinsonian symptoms and epileptic seizures when pharmaceutical drugs fail. In this paper, our goal is to design a desynchronization strategy for large networks of spiking neurons such that the neuronal activity of the network remains in the desynchronized regime for a long period of time after the removal of the stimulation. We develop a novel "Forced Temporal-Spike Time Stimulation (FTSTS)" strategy that harnesses the spike-timing dependent plasticity to control the synchronization of neural activity in the network by forcing the neurons in the network to artificially fire in a specific temporal pattern. Our strategy modulates the synaptic strengths of selective synapses to achieve a desired synchrony of neural activity in the network. Our simulation results show that the FTSTS strategy can effectively synchronize or desynchronize neural activity in large spiking neuron networks and keep them in the desired state for a long period of time after the removal of the external stimulation. Using simulations, we demonstrate the robustness of our strategy in desynchronizing neural activity of networks against uncertainties in the designed stimulation pulses and network parameters. Additionally, we show in simulation, how our strategy could be incorporated within the existing desynchronization strategies to improve their overall efficacy in desynchronizing large networks. Our proposed strategy provides complete control over the synchronization of neurons in large networks and can be used to either synchronize or desynchronize neural activity based on specific applications. Moreover, it can be incorporated within other desynchronization strategies to improve the efficacy of existing therapies for numerous neurological and psychiatric disorders associated with pathological synchronization.

Phase reduction and phase-based optimal control for biological systems: a tutorial

A powerful technique for the analysis of nonlinear oscillators is the rigorous reduction to phase models, with a single variable describing the phase of the oscillation with respect to some reference state. An analog to phase reduction has recently been proposed for systems with a stable fixed point, and phase reduction for periodic orbits has recently been extended to take into account transverse directions and higher-order terms. This tutorial gives a unified treatment of such phase reduction techniques and illustrates their use through mathematical and biological examples. It also covers the use of phase reduction for designing control algorithms which optimally change properties of the system, such as the phase of the oscillation. The control techniques are illustrated for example neural and cardiac systems.

Synchronization of heterogeneous oscillator populations in response to weak and strong coupling

Synchronous behavior of a population of chemical oscillators is analyzed in the presence of both weak and strong coupling. In each case, we derive upper bounds on the critical coupling strength which are valid for arbitrary populations of nonlinear, heterogeneous oscillators. For weak perturbations, infinitesimal phase response curves are used to characterize the response to coupling, and graph theoretical techniques are used to predict synchronization. In the strongly perturbed case, we observe a phase dependent perturbation threshold required to elicit an immediate spike and use this behavior for our analytical predictions. Resulting upper bounds on the critical coupling strength agree well with our experimental observations and numerical simulations. Furthermore, important system parameters which determine synchronization are different in the weak and strong coupling regimes. Our results point to new strategies by which limit cycle oscillators can be studied when the applied perturbations become strong enough to immediately reset the phase.

Hard real-time closed-loop electrophysiology with the Real-Time eXperiment Interface (RTXI)

The ability to experimentally perturb biological systems has traditionally been limited to static pre-programmed or operator-controlled protocols. In contrast, real-time control allows dynamic probing of biological systems with perturbations that are computed on-the-fly during experimentation. Real-time control applications for biological research are available; however, these systems are costly and often restrict the flexibility and customization of experimental protocols. The Real-Time eXperiment Interface (RTXI) is an open source software platform for achieving hard real-time data acquisition and closed-loop control in biological experiments while retaining the flexibility needed for experimental settings. RTXI has enabled users to implement complex custom closed-loop protocols in single cell, cell network, animal, and human electrophysiology studies. RTXI is also used as a free and open source, customizable electrophysiology platform in open-loop studies requiring online data acquisition, processing, and visualization. RTXI is easy to install, can be used with an extensive range of external experimentation and data acquisition hardware, and includes standard modules for implementing common electrophysiology protocols.

Initial comparison of energy measures for neural stimulation in a single conductance channel

This paper considers the utility of several alternative energy measures to reduce the energy required by a stimulation current source to charge a neuron membrane capacitance to a prescribed value in the case of a single sodium channel. For a simple case, minimizing the energy of the nonlinear channel conductance provides improved efficiency in terms of stimulator energy as compared to minimizing a squared-integral measure of the stimulation current. This work lays the foundation for expanding this investigation to a full conductance-based Hodgkin-Huxley model.

Isostable reduction of periodic orbits

The well-established method of phase reduction neglects information about a limit-cycle oscillator's approach towards its periodic orbit. Consequently, phase reduction suffers in practicality unless the magnitude of the Floquet multipliers of the underlying limit cycle are small in magnitude. By defining isostable coordinates of a periodic orbit, we present an augmentation to classical phase reduction which obviates this restriction on the Floquet multipliers. This framework allows for the study and understanding of periodic dynamics for which standard phase reduction alone is inadequate. Most notably, isostable reduction allows for a convenient and self-contained characterization of the dynamics near unstable periodic orbits.

Desynchronization of stochastically synchronized chemical oscillators

Experimental and theoretical studies are presented on the design of perturbations that enhance desynchronization in populations of oscillators that are synchronized by periodic entrainment. A phase reduction approach is used to determine optimal perturbation timing based upon experimentally measured phase response curves. The effectiveness of the perturbation waveforms is tested experimentally in populations of periodically and stochastically synchronized chemical oscillators. The relevance of the approach to therapeutic methods for disrupting phase coherence in groups of stochastically synchronized neuronal oscillators is discussed.

An energy-optimal approach for entrainment of uncertain circadian oscillators

We develop an approach to find an energy-optimal stimulus that entrains an ensemble of uncertain, uncoupled limit cycle oscillators. Furthermore, when entrainment occurs, the phase shift between oscillators is constrained to be less than a predetermined amount. This approach is illustrated for a model of Drosophila circadian activity, for which it performs better than a standard 24-h light-dark cycle. Because this method explicitly accounts for uncertainty in a given system and only requires information that is experimentally obtainable, it is well suited for experimental implementation and could ultimately represent what is believed to be a novel treatment for patients suffering from advanced/delayed sleep-phase syndrome.

Optimal entrainment of heterogeneous noisy neurons

We develop a methodology to design a stimulus optimized to entrain nonlinear, noisy limit cycle oscillators with uncertain properties. Conditions are derived which guarantee that the stimulus will entrain the oscillators despite these uncertainties. Using these conditions, we develop an energy optimal control strategy to design an efficient entraining stimulus and apply it to numerical models of noisy phase oscillators and to in vitro hippocampal neurons. In both instances, the optimal stimuli outperform other similar but suboptimal entraining stimuli. Because this control strategy explicitly accounts for both noise and inherent uncertainty of model parameters, it could have experimental relevance to neural circuits where robust spike timing plays an important role.

Future of seizure prediction and intervention: closing the loop

The ultimate goal of epilepsy therapies is to provide seizure control for all patients while eliminating side effects. Improved specificity of intervention through on-demand approaches may overcome many of the limitations of current intervention strategies. This article reviews the progress in seizure prediction and detection, potential new therapies to provide improved specificity, and devices to achieve these ends. Specifically, we discuss (1) potential signal modalities and algorithms for seizure detection and prediction, (2) closed-loop intervention approaches, and (3) hardware for implementing these algorithms and interventions. Seizure prediction and therapies maximize efficacy, whereas minimizing side effects through improved specificity may represent the future of epilepsy treatments.

Computation of reduced energy input current stimuli for neuron phase models

A regularly spiking neuron can be studied using a phase model. The effect of an input stimulus current on the phase time derivative is captured by a phase response curve. This paper adapts a technique that was previously applied to conductance-based models to discover optimal input stimulus currents for phase models. First, the neuron phase response θ(t) due to an input stimulus current i(t) is computed using a phase model. The resulting θ(t) is taken to be a reference phase r(t). Second, an optimal input stimulus current i(*)(t) is computed to minimize a weighted sum of the square-integral `energy' of i(*)(t) and the tracking error between the reference phase r(t) and the phase response due to i(*)(t). The balance between the conflicting requirements of energy and tracking error minimization is controlled by a single parameter. The generated optimal current i(*)t) is then compared to the input current i(t) which was used to generate the reference phase r(t). This technique was applied to two neuron phase models; in each case, the current i(*)(t) generates a phase response similar to the reference phase r(t), and the optimal current i(*)(t) has a lower `energy' than the square-integral of i(t). For constant i(t), the optimal current i(*)(t) need not be constant in time. In fact, i(*)(t) is large (possibly even larger than i(t)) for regions where the phase response curve indicates a stronger sensitivity to the input stimulus current, and smaller in regions of reduced sensitivity.